We present reduced-order models of unsteady low-Mach-number ideal gas flows in two-dimensional rectangular microchannels subject to first-order slip-boundary conditions. The pressure and density are related by a polytropic process, allowing for isothermal or isentropic flow assumptions. The Navier–Stokes equations are simplified using low-Mach-number expansions of the pressure and velocity fields. Up to first order, this approximation results in a system that is subject to no-slip condition at the solid boundary. The second-order system satisfies the slip-boundary conditions. The resulting equations and the subsequent pressure-flow-rate relationships enable modeling the flow using analog circuit components. The accuracy of the proposed models is investigated for steady and unsteady flows in a two-dimensional channel for different values of Mach and Knudsen numbers.

References

References
1.
Shan
,
X.
,
Wang
,
Z.
,
Jin
,
Y.
,
Wu
,
M.
,
Hua
,
J.
,
Wong
,
C.
, and
Maeda
,
R.
,
2005
, “
Studies on a Micro Combustor for Gas Turbine Engines
,”
J. Micromech. Microeng.
,
15
(
9
), p.
S215
S221
.10.1088/0960-1317/15/9/S07
2.
Isomura
,
K.
,
Murayama
,
M.
,
Teramoto
,
S.
,
Hikichi
,
K.
,
Endo
,
Y.
,
Togo
,
S.
, and
Tanaka
,
S.
,
2006
, “
Experimental Verification of the Feasibility of a 100 W Class Micro-scale Gas Turbine at an Impeller Diameter of 10 mm
,”
J. Micromech. Microeng.
,
16
(
9
), p.
S254
S261
.10.1088/0960-1317/16/9/S13
3.
Diab
,
N.
, and
Lakkis
,
I.
,
2012
, “
DSMC Simulations of Squeeze Film Between a Micro Beam Undergoing Large Amplitude Oscillations and a Substrate
,”
Proceedings of the ASME 2012 10th International Conference on Nanochannels, Microchannels, and Minichannels
, ICNMM2012.
4.
Senturia
,
S. D.
,
2001
,
Microsystem Design
,
Springer
,
New York
.
5.
Epstein
,
A.
,
2004
, “
Millimeter-Scale, Micro-electro-mechanical Systems Gas Turbine Engines
,”
ASME J. Eng. Gas Turbines Power
,
126
, p.
205
226
.10.1115/1.1739245
6.
Maxwell
,
J.
,
1879
, “
On Stresses in Rarified Gases Arising From Inequalities of Temperature
,”
Philos. Trans. R. Soc. London
,
170
, pp.
231
256
.10.1098/rspl.1878.0052
7.
Karniadakis
,
G.
,
Beskok
,
A.
, and
Aluru
,
N.
,
2005
,
Microflows and Nanoflows: Fundamentals and Simulation
, Vol.
29
,
Springer
,
New York
.
8.
Al-Bender
,
F.
,
Lampaert
,
V.
, and
Swevers
,
J.
,
2005
, “
The Generalized Maxwell-Slip Model: A Novel Model for Friction Simulation and Compensation
,”
IEEE Trans. Autom. Control
,
50
(
11
), pp.
1883
1887
.10.1109/TAC.2005.858676
9.
Cao
,
B.
,
Chen
,
G.
,
Li
,
Y.
, and
Yuan
,
Q.
,
2006
, “
Numerical Analysis of Isothermal Gaseous Flows in Microchannel
,”
Chem. Eng. Technol.
,
29
(
1
), pp.
66
71
.10.1002/ceat.200407079
10.
Arkilic
,
E.
,
Schmidt
,
M.
, and
Breuer
,
K.
,
1997
, “
Gaseous Slip Flow in Long Microchannels
,”
J. Microelectromech. Syst.
,
6
(
2
), pp.
167
178
.10.1109/84.585795
11.
Jang
,
J.
, and
Wereley
,
S.
,
2004
, “
Pressure Distributions of Gaseous Slip Flow in Straight and Uniform Rectangular Microchannels
,”
Microfluid. Nanofluid.
,
1
(
1
), pp.
41
51
.10.1007/s10404-004-0005-8
12.
Graur
,
I.
,
Meolans
,
J.
, and
Zeitoun
,
D.
,
2006
, “
Analytical and Numerical Description for Isothermal Gas Flows in Microchannels
,”
Microfluid. Nanofluid.
,
2
(
1
), pp.
64
77
.10.1007/s10404-005-0055-6
13.
Venerus
,
D.
, and
Bugajsky
,
D.
,
2010
, “
Compressible Laminar Flow in a Channel
,”
Phys. Fluids
,
22
, p.
046101
.10.1063/1.3371719
14.
Stevanovic
,
N.
,
2007
, “
A New Analytical Solution of Microchannel Gas Flow
,”
J. Micromech. Microeng.
,
17
(
8
), p.
1695
1702
.10.1088/0960-1317/17/8/036
15.
Qin
,
F.
,
Sun
,
D.
, and
Yin
,
X.
,
2007
, “
Perturbation Analysis on Gas Flow in a Straight Microchannel
,”
Phys. Fluids
,
19
, p.
027103
.10.1063/1.2564671
16.
Zohar
,
Y.
,
Lee
,
S.
,
Lee
,
W.
,
Jiang
,
L.
, and
Tong
,
P.
,
2002
, “
Subsonic Gas Flow in a Straight and Uniform Microchannel
,”
J. Fluid Mech.
,
472
(
1
), pp.
125
151
.10.1017/S0022112002002203
17.
Cai
,
C.
,
Sun
,
Q.
, and
Boyd
,
I.
,
2007
, “
Gas Flows in Microchannels and Microtubes
,”
J. Fluid Mech.
,
589
(
589
), pp.
305
314
.10.1017/S0022112007008178
18.
Lakkis
,
I.
,
2008
, “
System-Level Modeling of Microflows in Circular and Rectangular Channels
,” ICNMM2008, ASME.
19.
Issa
,
L.
, and
Lakkis
,
I.
,
2013
, “
Reduced Order Models of Low Mach Number Isothermal Flows in Microchannels
,”
Proceedings of ICNMM2013 the ASME 2013 11th International Conference on Nanochannels, Microchannels, and Minichannels
.
20.
Batchelor
,
G. K.
,
2000
,
An Introduction to Fluid Dynamics
,
Cambridge University
,
Cambridge, UK
.
21.
Majda
,
A.
, and
Lamb
,
K. G.
,
1991
,
Simplified Equations for Low Mach Number Combustion With Strong Heat Release
,
Springer
,
New York
.
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